New multi-criteria decision-making technique based on neutrosophic axiomatic design

There are several multicriteria decision-making (MCDM) approaches presented in the literature with their characteristics. Although traditional MCDM approaches are considered a proper implementation to select the best alternative from available types, they failed to consider uncertainty which is quite high and desires to be thoughtfully measured in the selection process. This research focuses on extending MCDM in the neutrosophic environment using axiomatic design (AD) as a novel contribution to selecting appropriate Computed Tomography (CT) devices. We present a new linguistic scale for evaluating criteria and alternatives based on single-valued triangular neutrosophic numbers (SVTrN). The proposed approach is superior to other existing approaches due to its simplicity and ability to simulate natural human thinking via considering truth, indeterminacy, and falsity degrees. Then, applying it will increase the value of imaging for medical decision-making and decrease needless costs. So, this study can be valuable to researchers by helping them consider the appropriate medical imaging system selection problem theoretically under uncertainty, and for governments and organizations to design better satisfying medical imaging evaluation systems.

• The traditional AD principles have been extended under various uncertain environments, but all existing AD approaches have limitations in managing inconsistent and indeterminate data. • The existing techniques employed in the literature for medical imaging system selection are not suitable for handling undefined, unknown, and unreliable information. • Several researchers applied a limited number of quantitative or qualitative criteria with their mathematical programming models for medical imaging system selection. • Almost existing approaches which extended AD in the fuzzy and intuitionistic fuzzy environment used a limited and inflexible linguistic scale that failed to consider indeterminacy and forced decision-makers to relate linguistic variables (LV) with fixed confirmation degrees. • There does not exist any study in literature which presented AD in neutrosophic environment for medical image selection, therefore, it is significant to introduce a new decision-making framework based on NAD.
The key contributions of this study are as follows: • A novel extended MCDM methodology depending on NAD principles is established to select appropriate CT devices. • A new linguistic scale for evaluating criteria and alternatives is presented.
• Our approach helps decision-makers to construct a logical and precise decision matrix based on a non-restrict confirmation degree with LV. • A descriptive study of CT medical image device choosing is presented to clear the usefulness and realism of the introduced method. • Evaluation with the present approach and sensitivity analysis are debated to explore the power and reliability of the gained decision outcomes. • Applying the proposed approach will increase the value of imaging for medical decision-making and decrease needless costs.
The remaining sections are divided as follows: "Preliminaries" section presents preliminaries of NS and AD. "Proposed approach for medical image modalities selection" section, presents the steps of the proposed approach. "Case study: results and analysis" section presents the application of the proposed approach for medical image modalities selection. "Sensitivity analysis" section provides the sensitivity analysis. "Comparative analysis" section illustrates the comparative study. "Managerial implications" section provides the managerial implications of this study. "Conclusions and future directions" section illustrates the conclusion and future directions of this work.

Preliminaries
In this part, some significant concepts of NS and AD principles are presented. 38 . Neutrosophic set (NS). Let ξ be the universe, and NS is D in ξ described by a T function T D , I function I D and a F function F D where T D , I D and F D are real standard elements of [0,1]. It can be represented as:

Neutrosophic concepts
There is no limitation on the sum of T D (x), I D (x), and F D (x) . So, Score function (SF) and accuracy function (AF). Is appropriate functions for comparing SVN. Assume ∼ D 1 = (T 1 , I 1 , F 1 ) be a SVN, then, the SF( ) of a SVNN are defined as follows: . Is a particular NS on the real number set R, whose T, I, F memberships are showed in Fig. 1, and represented as follows: Operations of SVTrN-number. If ∼ D 1 =< (m 1 , m 2 , m 3 ); T 1 , I 1 , F 1 > and ∼ D 2 =< (n 1 , n 2 , n 3 ); T 2 , I 2 , F 2 > is two SVTrN-number, then: SC and AF of SVTrN-number. The SF s( ∼ D 1 ) and AF a( ∼ D 1 ) can be defined as follows: Ranking of SVTrN-number. www.nature.com/scientificreports/ Axiomatic design principles. Independence axiom and information axiom are the most significant concepts of AD principles. The independence of functional requirements (FRs) that must be implemented is stated by the independence axiom. FRs indicate the smallest set of independent requirements which exemplifies the design goals. Where one of the most significant advantages of these methods is that if an alternative does not fulfil the FRs, the model stops it from being chosen as the best option.The information axiom declares that the design which has the minimum information content (IC) is the finest design between the designs that meet the independence axiom 39 . The information axiom is represented by the IC which is correlated to the probability of sustaining the plan goals. The IC i is given by: since Probability i is the probability of reaching a certain function requirement. . The logarithmic function is selected so that the IC will be additive when there are several FRs that must be fulfilled simultaneously. Where there are n FRs, the total IC is the total of all these probabilities 39 .
The probability of accomplishment is specified by the designer needs to attain regard to design range (DR) and the needs capability of the system regard to system range (SR). The intersection area of the DR and the SR is the common area where the satisfactory solution exists, as appears in Fig. 2.
In the case of the uniform probability distribution function Probability i can be presented as follows:

Proposed approach for medical image modalities selection
In the current section, the important concepts of AD in a neutrosophic environment are introduced. Also, a new MCDM approach for selecting appropriate medical image modalities is presented.
The extend of AD principles in neutrosophic environment. The values of criteria are presented using LV under the neutrosophic domain. Since we have imperfect information about SR and DR, then SR and DR for a specific criterion will be stated by utilizing "truth-membership (TM), " "falsity-membership (FM)" and "indeterminacy-membership (IM). " Therefore, the intersection areas of TM, IM, and FM functions of neutrosophic numbers can be attained as demonstrated in Fig. 3. The TM, IM, and FM of information content are represented as (IC T ) , ( IC I ) , and (IC F ) respectively which can be expressed as follows:  www.nature.com/scientificreports/ In NS-domain, the SF and the AF are used to compare neutrosophic values that can be expressed as s and a respectively. In this approach, we expand s and a with the IC in the AD environment. We represent SF i and AF i based on IC as represented in Eqs. (2) and (3).

The approach for MCDM problems based on AD principles and neutrosophic environment. If
. . , f rn is a group of FRs, that is the group of objectives for the criteria, ij ) ∈ P k is a preference value that is represented as SVTrN-number by the decision-maker, DM K ∈ DM for AT i ∈ AT regarding to C j ∈ C.
The proposed methodology of MCDM problems based on NAD for medical image modality selection are as follows: Step 1.  Table 1 for representation process of data.
Step 2. Aggregate decision-makers opinions using the average method via using Eq. (7) to get the summation of SVTrN-numbers and then divide it on their numbers for each criterion.
Step 3. Compute the IC T for each FR i .    www.nature.com/scientificreports/ where p ij1 and p ij3 are the L and U values of AT i by C i , where g j1 and g j3 are L and U values of FR i .
Step 4. Compute the IC I for each FR i .
where p ij1 and p ij3 are the L and U values of AT i by C i , where g j1 and g j3 are L and U values of FR i .
Step 5. Compute the IC F for each FR i .
where p ij1 and p ij3 are the L and U values of AT i by C i , where g j1 and g j3 are L and U values of FR i .
Step 6. Compute the value of SF for IC of AT i After computing the IC T ij , IC I ij , and IC F ij we get the form of SVN. So SF is computed based on Eq. (2).
Step 7. Compute the value of AF for IC of AT i .
After computing the IC T ij , IC I ij , and IC F ij we get the form of SVN. So AF is computed based on Eq. (3).
Based on ranking of SVTrN-number using Eqs. (12) and (13), choose the best alternative, according to SF i . In the condition that SF i of alternatives are equal, then rank them based on AT i . This part presented as follows: The flowchart of proposed approach presented in Fig. 4.

Case study: results and analysis
In our study, we aim to introduce a novel methodology for selecting the suitable medical image modality for a hospital involve 3000 employees. The selection process is based on five available devices of CT medical image modalities. The decision-makers in our case study are as follows: • D1: The main physician, • D2: A purchasing sector director, and • D3: A radiologist.
The decision-makers detected four criteria to select the available device as follows: • C1: Number of image slices, C2: Price, C3: Highest patient weight (kg), and C4: Excellence of After-sales service. Also, the FRs that must be assured by the CT device are presented in Table 2.
Truth-membership system design Truth-membership common area if p ij1 ≤ g j3 or p ij3 ≤ g j1 (21) IC I ij = 0 if p ij1 > g j3 or p ij3 > g j1 log 2 Indeterminacy-membership system design Indeterminacy-membership common area if p ij1 ≤ g j3 or p ij3 ≤ g j1 If SF i < SF j , then AT i < AT j (i.e.AT i is worse than AT j ) (26) If SF i = SF j and if , AF i < AF j , then AT i < AT j (i.e.AT i is worse than AT j ) AF i > AF j , then AT i > AT j (i.e.AT i is better than AT j ) AF i = AF j , then AT i = AT j (i.e.AT i is equal to AT j ) www.nature.com/scientificreports/ For solving this MCDM problem using the suggested approach apply the following steps: Step 1. Use LV presented in Table 1 to represent data. After then, construct decision matrices according to decision makers' selections for determining the significance rate of criteria as in Table 3.
Use the average method for aggregating weights of decision makers and make normalization after that as in Table 4. Finally, use the SF equation to compute weights values. Step 3.
Construct decision matrices for evaluating alternatives considering every criterion as in Table 5. www.nature.com/scientificreports/    www.nature.com/scientificreports/ Step 4. Make aggregation of decision makers' opinions about evaluating alternatives considering every criterion as in Table 6.
Compute the IC T using Eq. (20) as presented in Table 7.
Compute the IC I using Eq. (21) as presented in Table 8.
Compute the IC F using Eq. (22) as presented in Table 9.
Compute the SF of IC using Eq. (23) as appears in Table 10.
Compute the AF of IC using Eq. (24) as appears in Table 11.
Step 10. Ranking alternatives     www.nature.com/scientificreports/ Based on Eqs. (25) and (26) for ranking NS numbers. The final rank is as follows:

Sensitivity analysis
The sensitivity analysis of the suggested approach is conducted to assess the persistence of the priority rating and it can be an efficient way to determine the proposed approach's efficiency. A sensitivity analysis was performed on the attribute rank. So, we will show how various priorities of criteria will impact on final rank of alternatives. As we have 5 attributes, we get 24 case, so only 8 random cases has been shown in Fig. 5. Figure 5 show the change in the final rank of alternatives regarding to various priorities of criteria. As shown in the figure, changing the order of the attributes has a significant affect on the weights of the alternatives.
The result of our sensitivity analysis shows that alternative1 and 5 are the best two alternatives for CT device selection. Alternative 1 shows superior results in Fig. 5A, while alternative 5 shows superior ranking results in Fig. 5B. Alternatives 4 and 3 are the lowest rank of alternatives in all cases. While alternative 2 shows a medium rank in all results which indicates the stability during changes of weights.

Comparative analysis
In this part, we provide a comparison between the suggested approach and the other approach which presented AD in the intuitionistic fuzzy environment 27 .
By applying the proposed approach in 27 on our CT device selection problem, the final rank of alternatives are as follows: A1 > A2 > A5 > A4 > A3 . This rank is determine based on SF and AF values as in Eqs. (25) and www.nature.com/scientificreports/ (26). Where the highest SF value in our proposed is A5 and the highest SF value in 27 is A1. Table 12 summarizes the final rank of our CT selection problem using the approach presented in 27 and our suggested approach. For comparing ranks of the proposed approach with other approaches presented in 27 we used the following statistical methods as follows: Spearman's correlation. Calculates the linear correlation between two continuous variables. A correlation is linear when a change in one variable is correlated with a relative change in the other variable, that presented as follows: where ALT is the number of alternatives, and Distance n is the subtraction between alternatives ranking. The values of +1 or −1 indicate a strong correlation between two observations, and the 0 value indicates a low correlation.
Person's correlation. Calculate the strength of linear correlation. The values of +1 or −1 indicate a completely positive or negative linear correlation. and 0 value indicates unavailable linear correlation. Which represented as follows: where, cov(x, y) is the covariance between x, y , and σ x , σ y is the standard deviation of x and y respectively.
By calculating the Spearman correlation using Eq. (27) the correlation value is 0.7. Also, the Person correlation is 0.7, which indicates a strong correlation between the two approaches.
From a comparative study between our proposed approach and other presented approach in 27 , we concluded that our approach is simple to implement and more logical than the presented approach in 27 for the following reasons: • Since NS is more effective than IFS for dealing with uncertainty, then extending AD in a neutrosophic environment is more precise than AD which is presented in an intuitionistic fuzzy environment. • The extended AD in a neutrosophic environment simulates natural human thinking since indeterminacy degree does not depend on truth and falsity degree. Then, it can deal with situations in which fuzzy and intuitionistic fuzzy AD fails to handle. • Our proposed scale can deal with bigger areas in common ranges than presented in 27 .
• Finally, the scale presented in 27 , can't provide a logical confirmation degree since it is restricted to the linguistic variable. But our scale makes decision-makers feel free to use the suitable linguistic variable and its confirmation degree which can vary from one decision-maker to another.

Managerial implications
All existing hospitals need to select appropriate types of medical imaging systems which able to notice diseases at their beginning phase and then refining the patient's prediction intensely. As the selection process is a hard and complex task due to conflicting criteria and numerous available alternatives which exist nowadays, then we need a new extended MCDM approach. In this study, we presented for the first time a new extended MCDM approach based on NAD for handling uncertainty which exist usually in the selection process. The proposed approach proved its ability to deal with uncertainty and then make precise decisions. The proposed model can be a powerful guide for hospitals or medical organizations that desire to select appropriate medical imaging systems. Also, governments can use the suggested approach for making precise decisions about any social, economic, and environmental problems.

Conclusions and future directions
Lately, AD methods to decision making have grown in popularity. One of the most significant advantages of these methods is that if an alternative does not fulfil the FRs, the model stops it from being chosen as the best option. NAD techniques may handle both neutrosophic and crisp values simultaneously by extending the AD methodology to neutrosophic settings. This feature is not present in any other MCDM techniques described in www.nature.com/scientificreports/ the literature. In our study, we presented for the first time the principles of AD in a neutrosophic environment.